All of the 5th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$4.50$ each for students, and the group paid $$61.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$12.00$ each for students, and the group paid $$198.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4.5y = 61.5}$ ${34x+12y = 198}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-18y = -246}$ ${34x+12y = 198}$ Add the top and bottom equations together. $ -6y = -48 $ $ y = \dfrac{-48}{-6}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+4.5y = 61.5}$ to find $x$ ${8.5x + 4.5}{(8)}{= 61.5}$ $8.5x+36 = 61.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {34x+12y = 198}$ and get the same answer for $x$ ${34x + 12}{(8)}{= 198}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.